Poincaré developed topology and exploited this new
branch of mathematics in ingenious ways to study
the properties of differential equations.
Ideas and tools from this branch of mathematics
are particularly well suited to describe and to classify
a restricted but enormously rich class of chaotic
dynamical systems, and thus the term
chaos topology
refers to the description of such systems.
These systems correspond to flows that can be embedded
in three-dimensional spaces, but they are useful because
these are the only chaotic flows that can easily be visualized at present.

In this description there is a hierarchy
of structures that we study. This hierarchy can be
expressed in biological terms. The skeleton of the
attractor is its set of unstable periodic orbits,
the body is the branched manifold that describes the attractor,
and the skin that surrounds the attractor is the
surface of its bounding torus.

A deterministic trajectory from a prescribed initial condition
can exhibit chaotic behavior. In this case, once transients have died out, the trajectory
lies in a chaotic attractor.

A useful working definition of chaotic motion is motion that is:

deterministic

bounded

recurrent but not periodic

sensitive to initial conditions

Figure 1: A chaotic attractor which is produced by a folding mechanism.

Figure 2: A chaotic attractor which is produced by a tearing mechanism.

Figure 1 and Figure 2 show two examples of chaotic attractors.
What is shown is the motion of a point in the
phase space of the attractor. The trajectories shown
are solutions of a set of first order
nonlinear ordinary differential equations. This motion
is deterministic, bounded, and recurrent.

Sensitivity to initial conditions is shown in Figure 3.
In this figure two trajectories are shown that emerge
from nearby initial conditions for (a) the Rössler attractor
and (b) the Lorenz attractor. The initial
conditions differ by 0.000001 in the coordinate
plotted. For short times the two trajectories follow
each other so closely they cannot be distinguished, but
after sufficiently long times the trajectories are
wildly different. Long term prediction seems impossible.

Figure 3: Sensitivity to initial conditions for two different chaotic dynamics.

Chaotic attractors are built up by the endless repetition
of two fundamental processes: stretching and squeezing.
Stretching is responsible for sensitivity to initial
conditions and can be measured quantitatively by
positive Lyapunov exponents. Squeezing is responsible
for building up the layered structure, like a millefeuille.
The rate at which squeezing occurs is measured by the
negative Lyapunov exponents. The layered structure is
an example of a fractal structure. There are several notions of
fractal dimension, which take non-integer values in general.
In Figure 3 the Rössler attractor
is built up by repetition of stretching and folding.
The folded parts of the attractor are squeezed together
to build up a fractal. In Figure 3
the Lorenz attractor is built up by repetition of
tearing and squeezing. In this case the separated
parts of the attractor are squeezed together
to build up a fractal. Other attractors are built up by
combinations of these processes.

These two chaotic attractors look different. Is there
any way to show that they really are different? In
fact, what does different mean? The only sensible way
to define different is in a topological sense. More
precisely, two chaotic attractors are isotopic
(ambient isotopic to be precise, or equivalent for short)
if one can be deformed continuously into the
the other via a continuous deformation of the ambient space in which they live:
discontinuous processes such as tearing, gluing, or
surgery (tearing and gluing) are not allowed.
If two attractors are not equivalent
in this topological sense, they are different.
How to determine the equivalence or inequivalence of two
chaotic attractors is yet another miracle of topology.
Associated with (vulgar: in) every chaotic attractor there
are periodic orbits. There is an infinity of
such orbits; they are countable and are all unstable. This can be seen from
a simple intuitive argument. If you follow a chaotic
trajectory long enough it will come back to the neighborhood
of its initial position (Poincaré's
recurrent but not periodic property).
If you perturb the initial condition just the right way,
the perturbed trajectory will close up to form a periodic
orbit. This periodic orbit is not stable; otherwise the
initial condition would have relaxed to this orbit.

And now for the topological details.
In three dimensional space
an integer invariant can be associated to each pair of
closed orbits. This invariant is the Gauss Linking Number.
It can be defined by an integral

\(Lk(\vec{x},\vec{y}) = \frac{1}{4 \pi}\oint \oint
\frac{(\vec{x} - \vec{y}) \cdot d\vec{x} \wedge d\vec{y}}{|\vec{x}-\vec{y}|^3}\)
This integral expression is applied to two closed orbits
\(\vec{x}(s) = (x_1(s),x_2(s),x_3(s))\) and
\(\vec{y}(t) = (y_1(t),y_2(t),y_3(t))\) with
\(0 \le s,t \le 1\ ,\) \(\vec{x}(0)=\vec{x}(1)\ ,\) and
\(\vec{y}(0)=\vec{y}(1)\ ,\) with the condition that the two closed
orbits share no point in common, \(\vec{x}(s)\ne \vec{y}(t)\)
for any values of \(s,t\ .\) It is remarkable that this
integral, which appears to be of geometric origin, always
has integer values - which is a signature of topological
origins. This miracle can be explained many ways, all equivalent.
Each unstable periodic orbit is a knot
that is oriented by the direction of the
flow. Every oriented knot has associated
with it an oriented surface, called a Seifert
surface. The Gauss linking number
of two oriented knots is the sum of the
intersections, each carrying a sign \(\pm 1\ ,\)
of one knot with the Seifert surface of the
other. The sign is +1 at intersections where
the normal to the surface and tangent to the
knot are in the same direction (their dot product
is positive) and -1 otherwise.

In practice, it is often simpler to
compute \(Lk({\vec{x},\vec{y}})\) by projecting the two closed curves
onto a planar surface and counting the number of times their
projections cross. The crossings carry a sign, \(\pm 1\ ,\)
depending on which segment is closer to the observer
in the projection and the directions in which the projected
segments cross. The Gauss Linking Number is half the sum of the signed
crossings in the projection. This will be made explicit below.

The Gauss Linking Number between two orbits
remains unchanged while the orbits are disjoint: it can
only change when a segment of one orbit crosses through
a segment of the other orbit. At such points \( | \vec{x}-\vec{y}|=0 \)
and the integral fails to exist. Such crossings cannot occur for the
orbits in chaotic attractors because the crossing
point would then have two futures, one associated with
each periodic orbit. This violates determinism.

From each chaotic attractor we can extract a set of unstable
periodic orbits. In fact,
we can extract segments in a chaotic trajectory that are approximations to
some unstable periodic orbits. These segments are in the neighborhoods of unstable periodic
orbits that are typically of low period and low instability.
For each pair of these orbits we can compute
the Gauss Linking Number. To be equivalent, two chaotic
attractors must have not only a 1:1 correspondence between their
unstable periodic orbits, but the Linking Numbers of corresponding orbits
must be the same. This notion of equivalence is too restrictive
for applications in the physical sciences: we present a more
lenient definition at the end of Section III.

Figure 4: The Rössler attractor

Figure 5: First-return map of the Rössler attractor.

Figure 6: Link extracted from the Rössler attractor.

Figure 7: The Sprott D attractor

Figure 8: First-return map of the Sprott D attractor.

Figure 9: Link extracted from the Sprott D attractor.

We illustrate these ideas in Figs. from Figure 4 to Figure 9. These figures show
the projections of two chaotic attractors onto the
x-y planes in their respective phase spaces. Chaotic
behavior is produced by stretching and folding in both cases.
This is confirmed by the shape of the first return map
onto a Poincaré section for each of the attractors.

A Poincaré section is a surface that the trajectory
intersects each time around. Although it is not
always an easy matter to find such surfaces, in the
cases shown in Figure 4 and Figure 17 it is not overly difficult. For the
Rossler attractor the surface is the half plane parallel
to the \(x\)-\(z\) plane \( y = \) cst. with one edge going through the
hole in the middle of the attractor. For the other
attractor the Poincaré section is defined similarly.
The projections of the Poincaré sections onto the x-y planes
are indicated by the red line
segments in Figure 4 and Figure 17.

The intersection of the attractor with the Poincaré
section is a fractal. For these attractors the fractal
is very thin because the dissipation is large.
Both the \( x \) and \( z \) coordinates are recorded each
time the trajectory intersects the Poincaré section.
In the cases shown in Figure 5 and Figure 18 the first-return map is a plot of the
next value of \( x \) as a function of the previous value,
\( x_{n+1} \) vs.
\( x_n \ .\) Although these curves have a fractal structure,
the fractals are so thin they look like simple one-dimensional
curves with a maximum at M. The parabolic nature of the maximum of the
first return map is a signature that the mechanism that
produces chaotic behavior is stretching and folding, not
tearing and squeezing.

Segments of trajectories in the chaotic attractor that
are good approximations to unstable periodic orbits
can be located using first return maps. The sequence
of value \( x_i \) is searched to find the value of the index
\(k\) that minimizes \( |x_{k+p}-x_k| \ .\) The coordinate
values \( (x,y,z)_k \) are used as initial conditions
for a trajectory that comes very close to closing after
\( p \) times around the attractor. If this trajectory doesn't
almost close up before \( p \) periods, or \( p \) successive intersections
with the Poincaré section, the segment of
the chaotic trajectory starting
at \( (x,y,z)_k \) and ending at \( (x,y,z)_{k+p} \) is used as
an approximation for an unstable orbit
of period \( p \ .\) This is a simple way to locate unstable
periodic orbits in a chaotic attractor. The intersection
of the first return map with the diagonals \( x_{n+1}=x_n \)
in Figure 5 and Figure 18 easily locates the period one orbit(s).

Two unstable periodic orbits are plotted in Figure 6 and Figure 9
for each of the two attractors. Each orbit has been given
a name. The name is determined by the order in which the orbits
occur in the first return map. That is, each of the two branches
of the first return map has been given a name: 0 for the
ascending branch with slope >0 and 1 for the descending
branch with slope <0. The two branches are separated by
a maximum where the slope is zero. The assignment of a
discrete sequence of a small number of symbols to a continuous
trajectory is called symbolic dynamics. Symbol
sequences related by cyclic permutations describe the same
orbit: for example the symbol sequences 1011, 0111, 1110, and
1101 describe the four different initial conditions on a
Poincaré section for the same period-four orbit.

Linking numbers
are computed by counting signed crossings and dividing
by two. At each crossing the tangent vector at
the upper segment is rotated through the smallest angle
into the tangent vector to the lower segment. If the
rotation is right-handed the crossing is assigned the
index +1 ; if it is left-handed the crossing is assigned the
index -1. The Gauss Linking Number of the two orbits
is half the sum of the signed crossing indices.

The orbits labeled 10 and 1011 exhibit 6 negative crossings
for the Rossler attractor. In the Sprott D attractor
the two orbits with the same symbolic names have
8 negative crossings and two positive crossings.
In both cases, \(Lk(10,1011)=-3\ .\) The linking numbers for
three corresponding orbits in each of the two
attractors is shown in the Table below.

Linking numbers

Rössler

Sprott D

\( Lk(1,10) \)

-1

-1

\( Lk(1,1011) \)

-2

-2

\( Lk(10,1011) \)

-3

-3

Templates

Lists of orbits and tables of Linking Numbers are
enormously useful and rewarding. But they provide
a surplus of information and are, in some sense, overkill.
The information contained in such lists and tables
can be summarized very economically in a simple construction.
This looks like a cardboard model of the flow, of the type
used by Lorenz and Rössler to describe their chaotic attractors.
Although such models look very simple and intuitive, they have a
mathematical rigor provided by one of the few theorems in
this field that is useful to experimentalists, applied
mathematicians, and pure mathematicians alike. This theorem,
due to J. Birman and R. F. Williams, depends on a kind
of identification of different points that is more common in
algebra (for example: projections, homomorphisms) than topology.
Briefly, Birman and Williams identify two points
\( \vec{x} \) and \(\vec{y} \) in the phase space if the distance
between the futures of these points, \(\vec{x}(t) \)
and \( \vec{y}(t) \ ,\) decreases to zero as \(t \rightarrow \infty \ .\)
Such points are equivalent under the Birman-Williams
projection\[ \vec{x} \simeq \vec{y} \ .\]

Birman-Williams Theorem (1983)
Given a flow \(\phi_t\) on a three-dimensional manifold \(M^3\) having a hyperbolic structure on its chain recurrent set there is a knot holder \( (H, \overline{\phi}_t)\ ,\) \( H \subset M^3\ ,\) such that with perhaps one or two specified exceptions the periodic orbits under \(\phi_t\) correspond one-to-one to those under \(\overline{\phi}_t\ .\) On any finite subset of the periodic orbits the correspondence can be taken to be via isotopy.

\(
\lambda_1+\lambda_2+\lambda_3 < 0 ~~~(\mbox{dissipative condition})
\)
and produces a hyperbolic chaotic attractor. The
identification of all points with the same future
\( \vec{x} \simeq \vec{y} \) maps the chaotic attractor
onto a branched manifold and the flow in the chaotic
attractor to a semiflow on the branched manifold. The
periodic orbits in the chaotic attractor correspond in
a one-to-one way with the periodic orbits on the branched
manifold with one or two exceptions. For any finite
subset of periodic orbits the correspondence is an isotopy.

This identification of points has the effect
of projecting the attractor produced by the flow onto a
two dimensional set that is a manifold almost everywhere.
One of the two dimensions corresponds to the stretching direction
(\( \lambda_1 > 0 \)), the other to the flow direction
(\(\lambda_2 = 0\)). The projection direction corresponds
to the squeezing direction (\( \lambda_3 < 0 \)).
This structure is called a branched 2-manifold.
The organization of the unstable periodic orbits in the chaotic attractor
remains unchanged under this projection. This means that
the linking numbers of the orbits in the chaotic attractor can be computed
from the images of these orbits on the branched manifold.
This is relatively simple to do just by counting the
signed number of crossings. Further, the labeling of the branches
is used to label the orbits.

The branched manifold for the Rössler attractor is shown
on the left in Figure 10. The dark spots suggest the
location of the two fixed points of the
Rossler dynamical system. The projections of the orbits
10 (dashed line ) and 1011 (solid line) onto this
branched manifold are also shown. All crossings are
negative and occur in a localized region of this
branched manifold. The dashed line crosses the solid
lines 3 times in the branch labeled 1 and the dashed
lines cross the solid lines 3 times where the two
branches cross. The linking number is easily \( \frac{-3-3}{2}=-3 \ .\)

In Figure 11 we show two branched manifolds.
The one on the left describes the Sprott D attractor.
The one on the right is attempted simply by moving the
negative half-twist down the left hand side of the branched
manifold.

Figure 10: Branched manifold of Rössler attractor.

Figure 11: Branched manifold of Sprott D attractor.

If a branched manifold is useful for determining linking
numbers, the converse is also true. When a chaotic trajectory can be labeled
by a finite set of symbols, from a handful of
orbits identified within a chaotic attractor, linking
numbers of pairs of orbits can be used to determine the branched manifold
that describes the attractor: that is, the branched manifold
the attractor projects to under the Birman-Williams
identification. This insight has been applied to
experimental data. After a small number of orbits are
located in experimental data and their linking numbers
computed, an appropriate branched manifold can be identified.
Even better, this branched manifold predicts the linking
numbers of all other orbit pairs that exist in this model.
As a result, if additional orbits can be extracted from the data, the
branched manifold identification can be confirmed or rejected.
Since the branched manifold describes the mechanism that
creates the chaotic attractor, the topological indices - a
small set of integers - are sufficient to suggest the
mechanisms that produces chaos.

We return now to a notion of the equivalence of low-dimensional
chaotic attractors that is useful for physical applications.
We define two chaotic attractors to be equivalent if they
are described by branched manifolds that can be smoothly
deformed, one into the other. This notion of equivalence
is illustrated in Figure 10 and Figure 11.

The Birman-Williams theorem holds for flows in 3-d with a hyperbolic attractor. It
has most often been used to describe very dissipative dynamical systems:
those with \( | \lambda_3 | \gg \lambda_1\ .\) Attractors produced under
these conditions are thin fractals for which the projection to a branched manifold
requires little imagination. For this reason the theorem has sometimes been regarded
as a mathematical nicety of little real use. We emphasize here that the theorem holds
for dissipative systems with \( \lambda_1 \) only slightly smaller than
\( |\lambda_3| \ .\) For such systems the attractor is a very nebulous attractor
and it is not at all obvious either what the branched manifold looks like or how actually
to construct if from either data or simulations.

Bounding tori

Chaotic attractors produced by a three dimensional dynamical
system may have a spongy fractal structure and a fractal
dimension less than three, but they exist in a three
dimensional manifold. Another miracle of topology is that
the boundaries of all three-dimensional closed bounded surfaces
have been classified: all such boundaries are two
dimensional tori with \(g\) holes (\(g\) for genus).
The surface with \(g=0\)
is the ordinary sphere \(x^2+y^2+z^2=1\) in \(R^3\ ,\) the surface
with \(g=1\) is the tire tube, and so forth. If we can determine
three dimensional manifold with the smallest number of holes that contains a
chaotic attractor we can further refine the classification of
chaotic attractors by identifying the genus of the torus
bounding this manifold.

Figure 12: The Rössler attractor inside a genus-one bounding torus.

Such a three-manifold is determined by surrounding each point
in the chaotic attractor by a small ball of radius \(\epsilon \ .\)
The union of these balls is a three dimensional manifold.
Its boundary is a torus of genus \(g\ .\) The result is
that every three dimensional
chaotic attractor (that is, an attractor that exists in a
three-dimensional manifold) is bounded by a torus of genus
\(g=1\) or \(g \ge 3\ .\)

The Rössler attractor is of genus-one type.
The branched manifold that describes the Rössler
attractor is shown nestled inside a genus-one bounding
torus in Figure 12. Here it is obvious that the hole
in the torus excludes the focus at the center of
the attractor. In terms of this torus it is simple to
define a Poincaré surface of section. It is a disk
that is transverse to the flow inside the torus. The boundary
of the disk is a meridian in the surface of the torus.
The return map for the Rössler attractor, shown inFigure 5, describes how a point in the Poincaré section
is mapped back to the Poincaré section after a full
trip through the torus.

Figure 13: The Lorenz attractor inside a genus-three bounding torus.

The Lorenz attractor is of genus-three type.
Two holes exclude the symmetrically placed foci.
The third hole excludes the \(z\) axis.
The branched manifold that describes the Lorenz
attractor is shown nestled inside a genus-three bounding
torus in Figure 13. The three holes exclude the three
critical sets. The locations of the two foci are indicated
by small circles; the location of the regular saddle
on the \(z\) axis is indicated by a square. The
Poincaré global surface of section is also
simple to define once the bounding torus is known.
It consists of two disjoint disks (generally,
\(g-1\) disjoint disks for a genus-\(g\) torus, \(g>1\ ,\)
arranged so that the points in each disk flow to
exactly two disks in both the forward and reverse time
directions). The intersection of the Lorenz attractor
with each disk is a very thin fractal, almost one
dimensional. The return map for the Lorenz attractor
describes how initial conditions on the intersections
in each of the two components flow back to
these two components.

That the Rössler (and Sprott D) attractor exists inside a
genus-one bounding torus and the Lorenz attractor
exists inside a genus-3 bounding torus provides a very
simple demonstration that they belong to different
equivalence classes of chaotic attractors.

Four different templates

In this section we present four dynamical systems.
Each produces a chaotic attractor for
control parameter values within a certain
range. All four attractors are enclosed with a
genus-one bounding torus. For each, we present
the equations, specific control parameter values,
a figure, and a description of the mechanism
that produces chaotic behavior. The figure shows
a projection of the attractor onto a plane,
the Poincaré section chosen, the first return map,
and the branched manifold that the attractor
projects to.

Folding

The most popular example of the a simple folding
remains the Rössler system proposed in 1976\[ \dot{x} = -y-z \]

\( \dot{y} = x+ay \)

\( \dot{z} = b+z(x-c) \)

It is characterized by a first-return map with a
differentiable critical point separating the
increasing and decreasing branches. The Rössler
attractor is observed with parameter values
\(a=0.420\ ,\) \(b=2\) and \(c=4\ .\)

Figure 14: Folded chaos solution to the Rössler system.

Figure 15: First-return map of the Rössler attractor.

Figure 16: Template the Rössler attractor.

Inverted folding

One of the simplest chaotic attractors which
presents an inverted folding was proposed by J. C.
Sprott in his brute force search for simple quadratic
systems exhibiting chaotic behavior. Among his collection, system D

\( \dot{x} = -y \)

\( \dot{y} = x+z \)

\( \dot{z} = xz+ay^2 \)

is characterized by a unimodal map with a
differentiable maximum like the Rössler system but
has an additional half-turn applied to the two branches.
An attractor with an inverted folding
is shown with parameter value \(a=4.0\ .\)
Templates in Figs. 16 and 19 are topologically equivalent, that is, they are isotopic. But,
dynamically speaking, there is a big difference between these two systems, since the branch with the
half-turn is inside in Fig. 19 and outside in Fig. 16.

Figure 17: Inverted folded chaos solution to the Sprott D system.

Figure 18: First-return map of the Sprott D attractor.

Figure 19: Template the Sprott D attractor.

Tearing

It has been seen that the second possibility
to produce chaos was to conjugate tearing with squeezing.
When the corresponding attractor is embedded
within a genus-one bounding torus, the dynamics is still
governed by a unimodal map but with a critical
point which is not differentiable. In this case, the
first-return map to a Poincaré section has one
increasing branch and one decreasing branch separated
by a cusp. The return map is not differentiable at
this point. A set of equations for this type of
chaotic attractor was proposed by Rossler and
Ortoleva in 1978\[ \dot{x} = ax+by-cxy -\frac{(dz+e)x}{x+K_1} \]

Half-inverted tearing

It is also possible to construct a chaotic attractor
produced by a half-inverted tearing mechanism
conjugated with a squeezing mechanism. It is the solution
to another set of ordinary differential equations
proposed by Rössler in 1976\[ \dot{x} = x -xy -z \]

From these four proto-types, many other chaotic
attractors can be constructed as discussed in
Letellier et al (2005) and Gilmore & Letellier (2007).

The Universal Knot-Holder

When they introduced branched manifolds, Birman and Williams (1983) proposed that some knots might not exist
in the branched manifold for the Lorenz attractor. Later, Ghrist (1995) proved they were correct and in doing so, constructed a branched manifold that contains all tame knots. His knot-holder ( Figure 26) can rearranged to have the form shown in Figure 27.b.

Figure 26: Ghrist's universal knot-holder.

Figure 27: Rearrangement of the Ghrist's universal knot-holder.

Figure 28: Chaotic attractor solution to the flow containing representatives of every knots.

This later knot-holder
is contained within a genus-5 bounding torus. There is a corresponding flow governed by the equations - derived from a Chua circuit - reading as

\(\dot{x} = 7 (y-\phi) \)

\(\dot{y} = x-y+z \)

\(\dot{z} = - \beta y \)

where
\( \phi = \frac{2}{7}x-\frac{3}{14} (|x+1|-|x-1|) \, . \) A chaotic attractor solution to this system is shown in
Figure 28.

Examples from experimental data

Belousow-Zhabotinsky reaction

A beautiful set of experiments were carried out by
Swinney's group at the University of Texas at Austin
during the 1980s. a topological analysis was
carried out on data from one of these experiments by
Gilmore and his colleagues in 1991.
First, a set of
unstable periodic orbits were located in the
scalar time series. The set included about two dozen
orbits up to period 17.

The second step was the construction of a useful
embedding. A modified differential embedding
was constructed. A segment of the chaotic trajectory
in this embedding space is shown ion Figure 29.
Orbits up to period eight (nine orbits)
were mapped into this embedding space.
Their symbolic names were proposed by using the
first return map, also shown in Figure 30. The
pairwise linking numbers of these nine orbits
(\(9 \times 8/2=36\) of them) were
computed. The three linking numbers of the three lowest
period orbits, of periods 1, 2, and 3 were used to
guess an appropriate branched manifold with two
branches (Figure 36). This branched manifold was then used
to predict the remaining $33=36-3$ linking numbers
for the set of 9 orbits used for this analysis.
The predicted linking numbers agreed with the numbers
computed using the orbits extracted from the data.

Laser dynamics

Figure 32: Chaotic attractor observed in a CO\(_2\) laser with modulated losses (left) and a set of unstable periodic orbits extracted from it (right)

Thanks to a very fast dynamics and a high signal-to-noise ratio, lasers are model systems to study nonlinear dynamics and chaos . Among the
first systems whose topological structure was characterized are two
CO\(_2\) lasers operated in Pisa, Italy and Lille, France. Papoff
et al. (1992) studied the organization of periodic orbits in a
CO\(_2\) laser with saturable absorber at different values of control
parameters. They found that although the spectrum of periodic orbits
would change with control parameters, the template supporting them was
always a horseshoe template with zero global torsion. Lefranc and
Glorieux (1993) observed the very same template in data from a CO\(_2\)
laser with modulated losses. The very high signal to noise ratio of
this laser allowed them to extract beautiful experimental examples of
unstable periodic orbits which when superimposed provided a very good
approximation of the attractor (Fig. Figure 32). The self-linking
numbers of 27 orbits of period up to 12 as well as the linking numbers
between the 15 orbits of period 8 or less were computed and matched
that predicted by a Smale horseshoe template (Figure 36). Together with the
Belousov-Zhabotinsky analysis, this experiment has provided a clear
confirmation that knots and templates are not only theoretical
concepts but also powerful tools to characterize real systems.

Figure 33: From top to bottom: topological analysis of three chaotic attractors located in three different resonance tongues. From left to right: experimental bifurcation diagram in the tongue, attractor, return map and template. At lower modulation frequencies, templates display a higher global torsion.

Some of the first examples of different topological organizations were
given in other laser experiments. Using a pump-modulated fiber laser,
Boulant et al. (1997a) showed how the global torsion of the
template changes between the successive resonance tongues that are
encountered when scanning the modulation frequency ( Figure 33).

The twisted
horseshoe template (horseshoe template with added half-twist) was
characterized in a Nd:YAG laser by Boulant et al. (1997b) in
which they also reported a transition from twisted horseshoe to spiral
three-branch template to standard horseshoe template as a chaotic
resonance tongue was crossed (Boulant et al., 1998). These
three templates can in fact be considered as subtemplates of spiral
template with infinitely many branches. Very recently, beautiful
examples of spiral three-branch templates have been observed in fiber
laser experiments led by Used and Martin in Zaragoza, Spain (Used
et al., 2008).

Copper electrodissolution

A complete topological analysis of a copper electrodissolution experiment was achieved in 1995 by Letellier and co-workers in collaboration with Jack Hudson and Zihao Fei who made the experiments. Copper electrodissolution in H3PO4 has been found to undergo Hopf bifurcation to oscillatory behavior followed by period-doubling bifurcations to simple chaos. Such a chaos thus corresponds to a folded chaos and is characterized by a first-return map to a Poincaré section with two monotonic branches split by a maximum differentiable point. A set of three ordinary differential equations was obtained by a global modeling technique and the model was validated by showing that the
attractor it produces was schemed by the same template as the data were.

An incomplete list of open problems

Weakly dissipative 3D systems

In 1984, Lorenz proposed another 3D system. The most relevant departure from the so-called "Lorenz system" is that the newer one is weakly dissipative. This system is a set of three ordinary differential equations

\( \dot{x} = -y^2 -z^2 -ax + aF \)

\( \dot{y} = xy -bxz -y + G \)

\( \dot{z} = bxy +xz -z \)

This system was proposed as a model for the atmospheric circulation. When parameter values are \((a,b,F,G)=(0.25,4.0,8.0,1.0)\ ,\) a chaotic attractor is solution to this system. In spite of many attempts, the topology remains unknown although its Lyapunov dimension is less than 3.

Figure 37: The 3D weakly dissipative Lorenz attractor (1984).

Figure 38: First-return map of the 3D weakly dissipative attractor.

Figure 39: Branched manifold.

Hyperchaotic systems

In 1979, Rössler introduced a 4D system with two positive Lyapunov exponents and termed "hyperchaotic" such a system. The set of ordinary differential equations is

\( \dot{x} = -y -z \)

\( \dot{y} = x + 0.25 y + w \)

\( \dot{z} = 3.0 + xz \)

\( \dot{w} = -0.5 z + 0.05 w \)

and produces a hyperchaotic attractor when parameter values are \((a,b,c,d)=(0.25,3,0.5,0.05)\) and initial conditions are \((x_0,y_0,z_0,w_0)=(-10,-6,0,10.1)\ .\) The Lyapunov dimension is greater than 3 so the Birman-Williams theorem is not applicable to this system. In other words this chaotic attractor may not be described by a branched manifold. In fact, how to characterize the topology of this attractor is an open question although some attempts have been provided by Sciamarella and Mindlin, and by Lefranc.

Figure 40: The 4D hyperchaotic Rössler attractor (1979).

Figure 41: First-return map of the 4D hyperchaotic attractor.

Figure 42: Branched manifold.

Toroidal chaos

Recently a set of ordinary differential equations leading to a new type of chaotic
attractor was proposed by Li. This new type of attractor can be interpreted in terms of toroidal chaos. The Li system is

\( \dot{x}=a(y-x)+dxz \)

\( \dot{y}=bx+ky-xz \)

\( \dot{z}=cz+xy-ex^2 \)

With parameter values \(b=55\ ,\) \(c=11/6\ ,\) \(d=0.16\ ,\) \(e=0.65\) and \(k=20\ ,\) this sytem produces a toroidal chaotic attractor for which constructing a branched manifold is still an open question.

Figure 43: The 3D toroidal chaotic attractor (2007).

Figure 44: First-return map of the toroidal chaotic attractor.

Figure 45: Branched manifold.

Topology in higher dimension chaos

All simple closed curves in \(R^4\) are isotopic.
How then can we generalize the Birman-Williams theorem?
We have to consider separately its two main statements, namely (1) the
existence of a branched manifold, and (2) the 1-1 correspondance and
isotopy between periodic orbits in the attractor and
in the branched manifold, keeping in mind that in the end it is the
branched manifold that describes the geometric processes organizing
the attractor, and that periodic orbits are simply a tool
to reconstruct it from experimental data.

In fact, the Birman-Williams construction that projects an attractor
down to a branched manifold by squeezing phase space along the stable
manifold can be defined in any phase space dimension. If a
\(D\)-dimensional flow has \(d_u\) unstable directions then the projection
yields a \((d_u+1)\)-dimensional manifold with singularities. At
least when the stable manifold is one-dimensional (i.e., \(D=d_u+2\)),
it is easy to convince oneself that there is enough rigidity to build
a discrete classification of these manifolds (Gilmore and Lefranc,
2002). First, choose a section of the template, which is
\(d_u\)-dimensional. Then follow its deformations as one rotates around
the template. Because this section can be embedded as an hypersurface
of a \((D-1)\)-dimensional Poincaré section of the flow (\(D-1=d_u+1\)),
we can see that the net result over one turn will be
to fold the hypersurface onto itself (Fig. Figure 46).

Figure 46: A 2D section of the template for a 4D flow with 2 unstable directions evolves in 3D space before coming back to itself, creating cusp singularities in the process.

Besides the stacking order, additional information is contained in the
return map from the section into itself, which is a singular mapping
\(R^{d_u} \to R^{d_u}\ .\) The key idea is that the organization of its
singularities together with the stacking order provide signatures of
the stretching and squeezing mechanisms, and that the higher the space
dimension, the higher the order of these singularities. The return map
for the branched manifold of a 4D flow is a singular mapping \(R^2 \to
R^2\) and as such typically displays cusp singularities
(Fig. Figure 46). In fact, a template in 4D describes the
organization of a collection of cusps just as a template in 3D describes
the organization of a collection of folds.

However, branched manifolds live in an abstract space and we need a
tool to reconstruct them from data. An attractive idea is that knot
theory can be generalized to consider embeddings \(S^n \to S^{n+2}\) so
that in 4D it is possible to classify how two closed 2D surfaces are
intertwined. Furthermore, unstable periodic orbits are dressed with
invariant manifolds that are rigidly organized. Accordingly, Mindlin
and Solari (1997) have proposed to look at 2D surfaces contained in
the 3D invariant manifolds of periodic orbits and have found clear
signatures of invariant tori and Klein bottles in a 4D flow.

Another line of attack is that the principle of non-intersecting
trajectories that leads to knot theory in three dimensions is nothing
but a particular case of the more general principle of determinism. A
formulation of this principle that adapts to phase spaces of any
dimension is that although a volume element of phase space advected by
a chaotic flow may be stretched and squeezed, its exterior and
interior will remain disjoint, because its boundary cannot undergo
self-intersections. The technical statement is that an advected volume element preserves
its orientation, a dissipative version of the Liouville theorem. This
is to the principle of non-intersecting trajectories what Gauss's law
is to Maxwell equations.

Figure 47: Advection of a triangulation of periodic points from Poincaré section to Poincaré section.

It has been proposed to apply the orientation preservation principle
to the dynamics of simplicial spaces built on periodic points
(Fig. Figure 47). As periodic points move and some cells
get inverted, a transformation is applied that restores their
orientation (Lefranc, 2006). These transformations induce a
non-trivial dynamics in the simplicial space, which captures the
action of the stretching and squeezing mechanisms on the original
flow. For 3D flows, this formalism appears to be equivalent to the
knot-theoretical approach, but its higher-dimensional version remains
to be built. The relevance of simplicial spaces for topological
analysis had been first advocated by Sciamarella and Mindlin
(1999,2001), who proposed to use homology theory to explore the
structure of chaotic attractors in three or more dimensions.

Historical background

Among those who contributed to chaos topology,
the first one who made a link between dynamical
systems (or differential equations) and topology
was Henri Poincaré. He appropriated a name,
Analysis Situs, to describe his approach.
This name was coined by Gottfried Leibniz (1646-1716)
to describe a new approach to geometry depending
on context (situation). The term topology was first used by
Johann Listing (1808-1882) in his book
entitled Vorstudien zur Topologie (1847).
The first problem solved using a
topological approach was proposed by the students
at Konigsberg and resolved by Leonhard Euler (1707-1783)
in 1736. This is the famous
Seven bridges of Königsberg walking tour problem.
Poincaré came "naturally" to analysis situs
(qualitative analysis) when he realized that the
three-body problem does not have any general analytical
solution. A possible alternative was to investigate
a trajectory in the phase space. He then used the
concepts of singular (or fixed) points, Poincaré
section and bifurcations. Poincaré also inherited
the use of periodic orbits as a breach in an affordable place
from George William Hill
who introduced them in his 1878 Lunar theory. Indeed,
Poincaré understood that

Given a set of equations ... and any particular solution of these equations, one can always find a periodic solution (of course, the period might be very long), such that the difference between these two solutions is as small as you would like, for as long as you would like. So what makes the periodic solutions so valuable to us is that they are the only keyhole through which we might get a glimpse of things that have been beyond reach up to this time.

Another major attempt was provided by Edward
Lorenz in his 1963 paper by using isopleths drawn on a
surface and his first-return map to maxima
of the \(z\)-time series. But branched manifolds
were first drawn under the suggestive names of
paper model or origami by Rössler in 1976.
And some of the original drawings do not differ
too much from the recent templates, as seen for instance
with the blender ( Figure 48 and Rössler, 1976) or the
paper-sheet model published in 1977 by Rössler. A more formal geometric description
of the Lorenz attractors was proposed by Guckenheimer and Williams (1979). Knot-holders
were first introduced by Birman and Williams (1983).

Figure 49: Picture used by Andronov et al to described the dynamics of the universal circuit.

Rössler inheritated his qualitative approach
from Andronov, Khaikin and Vitt's book entitled
Theory of oscillators, published in 1937.
For instance, the so-called universal circuit
was described using an S-shaped surface with
a switching mechanism ( Figure 48).
Such a qualitative approach deeply impressed
Rössler, who started to understand the structure of
chaotic attractors by modeling them using
stretched and folded ribbon. Quickly, Rössler made
the link between the attractor and the first-return
map, mainly because it was the simplest way for him
to prove the existence of a chaotic attractor
using the Li-Yorke theorem period-three implies chaos.
Hereafter, Rössler used paper-sheet models
and maps - drawn by hand - to construct different types
of chaotic attractors, as reviewed in Letellier et al (2006).

Remarks

While our examples have been limited for clarity to strongly dissipative systems, a template can be constructed for any attractor fitting into a three-dimensional phase space. Indeed, knots are insensitive to dissipation and templates can be reconstructed from the knot invariants alone, regardless of whether a symbolic coding is available or not to label the orbits. As a matter of fact, it has been shown that template analysis can be used to extract symbolic dynamical information from orbits of a weakly dissipative attractor and construct a symbolic coding of it (Lefranc et al., 1994; Plumecoq and Lefranc, 2000).